RENDICONTI DEL SEMINARIO MATEMATICO

Periodic points and chaotic dynamics 125the recent years by Kennedy, Yorke and their collaborators in a series of fundamentalpapers in this area (see [33, 35, 36, 37, 38]). Our goal instead is to take advantage ofour simplified framework in which we consider only sets which are homeomorphic toa square and prove the existence of periodic points of any order. To this aim, we haveto apply Theorem 1 to the iterates of the map ψ and select carefully some subset ofthe domain D in order to find “true” periodic points (for instance those with a longminimal period). First, however, we need a further definition, taken from [39].We say that ψ : X ⊇ D ψ → X has a chaotic dynamics of coin-tossing typeon k symbols if k ≥ 2 and there is a metrizable space Z ⊆ X and k pairwise disjointcompact sets W 1 ,... , W k ⊆ Z ∩ D ψ such that, for each two-sided sequence (s n ) n∈Zwiths n ∈ {1,...,k}, ∀ n ∈ Z,there is a sequence of points (z n ) n∈Z withz n ∈ W sn and z n+1 = ψ(z n ), ∀ n ∈ Z.In other words, any possible itinerary on the sets W 1 ..., W k is followed by somepoint.As an auxiliary tool in order to obtain at the same time the existence of suchkind of chaotic trajectories and also the fact that all the periodic itineraries can befollowed by some periodic point, we have the following theorem (see [59]), which isalso reminiscent of some results in [34] and [73].THEOREM 3. Assume that there is a (double) sequence of oriented rectangles( k ) k∈Z and maps ((D k ,ψ k )) k∈Z , with D k ⊆ A k , such that (D k ,ψ k ) :  k ⊳ k+1for each k ∈ Z. Then the following conclusions hold:(a 1 ) There is a sequence (w k ) k∈Z with w k ∈ D k and ψ k (w k ) = w k+1 for all k ∈ Z;(a 2 ) For each j ∈ Z there is a compact and connected set C j ⊆ D j satisfyingC j ∩ (A j ) + b ̸= ∅, C j ∩ (A j ) + t ̸= ∅and such that for each w ∈ C j there is a sequence (y l ) l≥ j , with y l ∈ D l andy j = w, y l+1 = ψ l (y l ) for each l ≥ j;(a 3 ) If there are integers h, k with h < k such that  h =  k , then there is a finitesequence (z i ) h≤i≤k , with z i ∈ D i and ψ i (z i ) = z i+1 for each i = h,...,k − 1,such that z h = z k , that is, z h is a fixed point of ψ k−1 ◦ ··· ◦ ψ h .The proof of (a 1 ) and partially also that of (a 2 ) could be given by adapting toour setting the argument in [33, Lemma 3 and Proposition 5]. As to (a 3 ), we applyTheorem 1. The existence of the continuum (compact connected set) C j in (a 2 ) is abyproduct of the topological lemma that we employ also in the proof of Theorem 1.We give all the main details along the proof of Theorem 11 in Section 3.2 and refer to[59] for more information.